If the laminate has a rounded front edge, or a round joint at the backsplash, it's going to be quite difficult to cover. It is possible to glue laminate on top of laminate, but that's not a very good idea if the base laminate is coming loose, or bubbled up anywhere, or has a rounded joint.

I would look under the countertop to see how it's fastened down. In years gone by they nailed down the plywood base, then glued laminate to it. That's how our kitchen was done, and it's hard to rip up the countertop without ruining the cabinets. (If it was me, I might give it a very careful try anyway.)

OTOH if the countertop was screwed down from below, unscrewing the screws may make it possible to remove the old and install a new one. A new laminate countertop is probably the cheapest fix.

It *might* be possible to cover the countertop with epoxy as Re-design says, but there are all sorts of practical issues. One gallon of epoxy (about $80) will only cover about six feet of countertop 1/8" thick... IF the countertop is absolutely level. If it's not perfectly level all over, you're screwed from the get-go.

I painted our countertops four years ago, and it held up very well. I'm going to repaint them soon just to cover the chips. If you want exact steps, check out a good tutorial... but here's what I did (I should write an Instructable about this): sand the formica lightly just to roughen the surface. Clean them with a good degreaser. Apply a coat of good quality primer. Paint with a regular or textured paint.... I used Rustoleum's Hammered Black Metal spray. Note: it's now available in quart cans; I used spray and will never do that again as we found overspray in the oddest places for months. Do at least three coats of paint. Seal with at least two coats of clear lacquer. Let everything cure for a few days before you set anything on the counters. Be prepared to do without your counters for at least a week, which is harder than you'd think! But after it was done, it looked fantastic! People thought we'd gotten new marble countertops!

Ive seen ads for companies that simply lay a new countertop over the old one. Usually it's some sort of synthetic material, but some even offer artificial granite which looks pretty nice. Not really a DIY project though.

You could get a counter top replacement company out to give you a bid. You might be surprised at how economical that can be. It's impossible to just recover the counter top with new formica unless it is a very simple counter top. You could try tiling over it but you would have problems with the tiles coming up and problems getting the front of the counter right. I don't know of any paint that would work. There are some colored epoxies that are poured in a thick layer used for restaurant table tops that could be used but that would be likely as expensive as new counter tops. There are stone tops that are very durable, look great but a pretty pricy. I'm outta ideas but maybe someone else has some better news. I still think my first idea is good shot.